Part of a model is visible while the rest is invisible. Glitch?

This is just a guess, but I had the same problem when I was starting out with Blender:

Meshes only have one side.

A normal cube or sphere has a single 'outside' pointing out to the viewer. If you make a 2-dimensional model (like a sheet of paper) then you will only be able see it from one side. If you don't close an object correctly in blender, then when you import it, it won't have a single continuous 'outside.'

The best analogy I can think of is a 'ghost costume' made of a sheet: if you think of the sheet of paper that's only opaque from one side, if you make that sheet into a shape, still if you look at it from below you'll see right through the whole thing

A brute-force check of this is to 'solidify' your model in Blender then import it again (save a copy as this will doubtless destroy your carefully made model): http://wiki.blender.org/index.php/Doc:2.6/Manual/Modeling/Meshes/Editing/Faces

When "the rest of it disappears too", do you mean the same side that is invisible from the front view? your normals may be backwards. Go into blender and try flipping the normals, or the whole side that is invisible. If that doesn't work, post back what you find out.